# Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

• Interpret a phase diagram
• State Dalton’s law
• Identify and describe the triple point of a gas from its phase diagram
• Describe the state of equilibrium between a liquid and a gas, a liquid and a solid, and a gas and a solid

Up to now, we have considered the behavior of ideal gases. Real gases are like ideal gases at high temperatures. At lower temperatures, however, the interactions between the molecules and their volumes cannot be ignored. The molecules are very closecondensation occursand there is a dramatic decrease in volume, as seen in Figure 13.28. The substance changes from a gas to a liquid. When a liquid is cooled to even lower temperatures, it becomes a solid. The volume never reaches zero because of the finite volume of the molecules.

Figure 13.28 A sketch of volume versus temperature for a real gas at constant pressure. The linearstraight linepart of the graph represents ideal gas behavior—volume and temperature are directly and positively related and the line extrapolates to zero volume at $–273.15ºC–273.15ºC size 12{ +- "273" "." "15"°C} {}$, or absolute zero. When the gas becomes a liquid, however, the volume actually decreases precipitously at the liquefaction point. The volume decreases slightly once the substance is solid, but it never becomes zero.

High pressure may also cause a gas to change phase to a liquid. Carbon dioxide, for example, is a gas at room temperature and atmospheric pressure, but becomes a liquid under sufficiently high pressure. If the pressure is reduced, the temperature drops and the liquid carbon dioxide solidifies into a snow-like substance at the temperature $–78ºC–78ºC size 12{ +- "78"°C} {}$. Solid $CO2CO2 size 12{"CO" rSub { size 8{2} } } {}$ is called dry ice. Another example of a gas that can be in a liquid phase is liquid nitrogen $(LN2)(LN2) size 12{ $$"LN" rSub { size 8{2} }$$ } {}$. $LN2LN2 size 12{"LN" rSub { size 8{2} } } {}$ is made by liquefaction of atmospheric airthrough compression and cooling. It boils at 77 K $(–196ºC)(–196ºC) size 12{ $$–"196"°C$$ } {}$ at atmospheric pressure. $LN2LN2 size 12{"LN" rSub { size 8{2} } } {}$ is useful as a refrigerant and allows for the preservation of blood, sperm, and other biological materials. It is also used to reduce noise in electronic sensors and equipment, and to help cool down their current-carrying wires. In dermatology, $LN2LN2 size 12{"LN" rSub { size 8{2} } } {}$ is used to freeze and painlessly remove warts and other growths from the skin.

# PV Diagrams

### PV Diagrams

We can examine aspects of the behavior of a substance by plotting a graph of pressure versus volume, called a PV diagram. When the substance behaves like an ideal gas, the ideal gas law describes the relationship between its pressure and volume. That is

13.68 $PV=NkT(ideal gas).PV=NkT(ideal gas). size 12{ ital "PV"= ital "NkT" $$"ideal gas"$$ "." } {}$

Now, assuming the number of molecules and the temperature are fixed,

13.69 $PV=constant(ideal gas, constant temperature).PV=constant(ideal gas, constant temperature). size 12{ size 11{ ital "PV"="constant" $$"ideal gas, constant temperature"$$ "." }} {}$

For example, the volume of the gas will decrease as the pressure increases. If you plot the relationship $PV=constantPV=constant size 12{ size 11{ ital "PV"="constant"}} {}$ on a $PVPV size 12{ ital "PV"} {}$ diagram, you find a hyperbola. Figure 13.29 shows a graph of pressure versus volume. The hyperbolas represent ideal-gas behavior at various fixed temperatures, and are called isotherms. At lower temperatures, the curves begin to look less like hyperbolas—the gas is not behaving ideally and may even contain liquid. There is a critical point—that is, a critical temperature—above which liquid cannot exist. At sufficiently high pressure above the critical point, the gas will have the density of a liquid but will not condense. Carbon dioxide, for example, cannot be liquefied at a temperature above $31ºC31ºC size 12{"31" "." 0°C} {}$. Critical pressure is the minimum pressure needed for liquid to exist at the critical temperature. Table 13.3 lists representative critical temperatures and pressures.

Figure 13.29 $PVPV size 12{ ital "PV"} {}$ diagrams. (a) Each curve—isotherm—represents the relationship between $PP size 12{P} {}$ and $VV size 12{V} {}$ at a fixed temperature; the upper curves are at higher temperatures. The lower curves are not hyperbolas, because the gas is no longer an ideal gas. (b) An expanded portion of the $PVPV size 12{ ital "PV"} {}$ diagram for low temperatures, where the phase can change from a gas to a liquid. The term vapor refers to the gas phase when it exists at a temperature below the boiling temperature.
Substance Critical Temperature Critical Pressure
$KK size 12{K} {}$ $ºCºC size 12{°C} {}$ $PaPa size 12{"Pa"} {}$ $atmatm size 12{"atm"} {}$
Water 647.4 374.3 $22.12×10622.12×106 size 12{"22" "." "12"×"10" rSup { size 8{6} } } {}$ 219.0
Sulfur dioxide 430.7 157.6 $7.88×1067.88×106 size 12{7 "." "88" times "10" rSup { size 8{6} } } {}$ 78.0
Ammonia 405.5 132.4 $11.28×10611.28×106 size 12{"11" "." "28"×"10" rSup { size 8{6} } } {}$ 111.7
Carbon dioxide 304.2 31.1 $7.39×1067.39×106 size 12{7 "." "39"×"10" rSup { size 8{6} } } {}$ 73.2
Oxygen 154.8 −118.4 $5.08×1065.08×106 size 12{5 "." "08"×"10" rSup { size 8{6} } } {}$ 50.3
Nitrogen 126.2 −146.9 $3.39×1063.39×106 size 12{3 "." "39"×"10" rSup { size 8{6} } } {}$ 33.6
Hydrogen 33.3 −239.9 $1.30×1061.30×106 size 12{1 "." "30"×"10" rSup { size 8{6} } } {}$ 12.9
Helium 5.3 −267.9 $0.229×1060.229×106 size 12{0 "." "229" times "10" rSup { size 8{6} } } {}$ 2.27
Table 13.3 Critical Temperatures and Pressures

# Phase Diagrams

### Phase Diagrams

The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so $PTPT size 12{ ital "PT"} {}$ graphs are called phase diagrams. Figure 13.30 shows the phase diagram for water. Using the graph, if you know the pressure and temperature you can determine the phase of water. The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist, that is, they exist together in ratios, depending on pressure and temperature. For example, the boiling point of water is $100ºC100ºC size 12{"100"°C} {}$ at 1.00 atm. As the pressure increases, the boiling temperature rises steadily to $374ºC374ºC size 12{"374"°C} {}$ at a pressure of 218 atm. A pressure cooker—or even a covered pot—will cook food faster because the water can exist as a liquid at temperatures greater than $100ºC100ºC size 12{"100"°C} {}$ without all boiling away. The curve ends at a point called the critical point, because at higher temperatures the liquid phase does not exist at any pressure. The critical point occurs at the critical temperature, as you can see for water from Table 13.3. The critical temperature for oxygen is $–118ºC–118ºC size 12{ +- "118"°C} {}$, so oxygen cannot be liquefied above this temperature.

Figure 13.30 The phase diagram ($PTPT size 12{ ital "PT"} {}$ graph) for water. Note that the axes are nonlinear and the graph is not to scale. This graph is simplified—there are several other exotic phases of ice at higher pressures.

Similarly, the curve between the solid and liquid regions in Figure 13.30 gives the melting temperature at various pressures. For example, the melting point is $0ºC0ºC size 12{0°C} {}$ at 1.00 atm, as expected. Note that, at a fixed temperature, you can change the phase from solid (ice) to liquid (water) by increasing the pressure. Ice melts from pressure in the hands of a snowball maker. From the phase diagram, we can also say that the melting temperature of ice rises with increased pressure. When a car is driven over snow, the increased pressure from the tires melts the snowflakes; afterwards, the water refreezes and forms an ice layer.

At sufficiently low pressures, there is no liquid phase, but the substance can exist as either gas or solid. For water, there is no liquid phase at pressures below 0.00600 atm. The phase change from solid to gas is called sublimation. It accounts for large losses of snow pack that never make it into a river, the routine automatic defrosting of a freezer, and the freeze-drying process applied to many foods. Carbon dioxide, on the other hand, sublimates at standard atmospheric pressure of one atm. The solid form of $CO2CO2 size 12{"CO" rSub { size 8{2} } } {}$ is known as dry ice because it does not melt. Instead, it moves directly from the solid to the gas state.

All three curves on the phase diagram meet at a single point, the triple point, where all three phases exist in equilibrium. For water, the triple point occurs at 273.16 K $(0.01ºC)(0.01ºC) size 12{ $$0 "." "01"°C$$ } {}$, and is a more accurate calibration temperature than the melting point of water at one atm, or 273.15 K $(0ºC)(0ºC) size 12{ $$0 "." 0°C$$ } {}$. See Table 13.4 for the triple point values of other substances.

# Equilibrium

### Equilibrium

Liquid and gas phases are in equilibrium at the boiling temperature (see Figure 13.31). If a substance is in a closed container at the boiling point, then the liquid is boiling and the gas is condensing at the same rate without net change in their relative amount. Molecules in the liquid escape as a gas at the same rate at which gas molecules stick to the liquid, or form droplets and become part of the liquid phase. The combination of temperature and pressure has to be just right; if the temperature and pressure are increased, equilibrium is maintained by the same increase of boiling and condensation rates.

Figure 13.31 Equilibrium between liquid and gas at two different boiling points inside a closed container. (a) The rates of boiling and condensation are equal at this combination of temperature and pressure, so the liquid and gas phases are in equilibrium. (b) At a higher temperature, the boiling rate is faster and the rates at which molecules leave the liquid and enter the gas are also faster. Because there are more molecules in the gas, the gas pressure is higher and the rate at which gas molecules condense and enter the liquid is faster. As a result the gas and liquid are in equilibrium at this higher temperature.
Substance Temperature Pressure
$KK size 12{K} {}$ $ºCºC size 12{°C} {}$ $PaPa size 12{"Pa"} {}$ $atmatm size 12{"atm"} {}$
Water 273.16 0.01 $6.10×1026.10×102 size 12{6 "." "10"×"10" rSup { size 8{2} } } {}$ 0.00600
Carbon dioxide 216.55 −56.60 $5.16×1055.16×105 size 12{5 "." "16" times "10" rSup { size 8{5} } } {}$ 5.11
Sulfur dioxide 197.68 −75.47 $1.67×1031.67×103 size 12{1 "." "67"×"10" rSup { size 8{3} } } {}$ 0.0167
Ammonia 195.40 −77.75 $6.06×1036.06×103 size 12{6 "." "06"×"10" rSup { size 8{3} } } {}$ 0.0600
Nitrogen 63.18 −210.0 $1.25×1041.25×104 size 12{1 "." "25"×"10" rSup { size 8{4} } } {}$ 0.124
Oxygen 54.36 −218.8 $1.52×1021.52×102 size 12{1 "." "52" times "10" rSup { size 8{2} } } {}$ 0.00151
Hydrogen 13.84 −259.3 $7.04×1037.04×103 size 12{7 "." "04"×"10" rSup { size 8{3} } } {}$ 0.0697
Table 13.4 Triple Point Temperatures and Pressures

One example of equilibrium between liquid and gas is that of water and steam at $100ºC100ºC size 12{"100"°C} {}$ and 1.00 atm. This temperature is the boiling point at that pressure, so they should exist in equilibrium. Why does an open pot of water at $100ºC100ºC size 12{"100"°C} {}$ boil completely away? The gas surrounding an open pot is not pure water: It is mixed with air. If pure water and steam were in a closed container at $100ºC100ºC size 12{"100"°C} {}$ and 1.00 atm, they would coexist—but with air over the pot, there are fewer water molecules to condense, and water boils. What about water at $20ºC20ºC size 12{"20" "." 0°C} {}$ and one atm? This temperature and pressure correspond to the liquid region, yet an open glass of water at this temperature will completely evaporate. Again, the gas around it is air and not pure water vapor, so that the reduced evaporation rate is greater than the condensation rate of water from dry air. If the glass is sealed, then the liquid phase remains. We call the gas phase a vapor when it exists, as it does for water at $20ºC20ºC size 12{"20" "." 0°C} {}$, at a temperature below the boiling temperature.

Explain why a cup of water—or soda—with ice cubes stays at $0ºC,0ºC, size 12{0°C} {}$ even on a hot summer day.

#### Solution

The ice and liquid water are in thermal equilibrium, so that the temperature stays at the freezing temperature as long as ice remains in the liquid. Once all of the ice melts, the water temperature will start to rise.

# Vapor Pressure, Partial Pressure, and Dalton’s Law

### Vapor Pressure, Partial Pressure, and Dalton’s Law

Vapor pressure is defined as the pressure at which a gas coexists with its solid or liquid phase. Vapor pressure is created by faster molecules that break away from the liquid or solid and enter the gas phase. The vapor pressure of a substance depends on both the substance and its temperature—an increase in temperature increases the vapor pressure.

Partial pressure is defined as the pressure a gas would create if it occupied the total volume available. In a mixture of gases, the total pressure is the sum of partial pressures of the component gases, assuming ideal gas behavior and no chemical reactions between the components. This law is known as Dalton’s law of partial pressures, after the English scientist John Dalton (1766–1844), who proposed it. Dalton’s law is based on kinetic theory, where each gas creates its pressure by molecular collisions, independent of other gases present. It is consistent with the fact that pressures add according to Pascal’s Principle. Thus water evaporates and ice sublimates when their vapor pressures exceed the partial pressure of water vapor in the surrounding mixture of gases. If their vapor pressures are less than the partial pressure of water vapor in the surrounding gas, liquid droplets or ice crystals—frost—form.